Exchangeable and partially exchangeable random partitions

Abstract

Call a random partition of the positive integers partially exchangeable if for each finite sequence of positive integers n1,..., nk, the probability that the partition breaks the first n1+...+nk integers into k particular classes, of sizes n1,...,nk in order of their first elements, has the same value p(n1,...,nk) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric function p(n1,...nk). A representation is given for partially exchangeable random partitions which provides a useful variation of Kingman's representation in the exchangeable case. Results are illustrated by the two-parameter generalization of Ewens' partition structure. © 1995 Springer-Verlag.